1. Field of the Invention
The present invention relates to a thermal analysis instrument, and more particularly, to a power compensation differential scanning calorimeter.
2. Background of the Invention
Differential scanning calorimeters (DSCs) measure the flow of heat to a sample to be analyzed as a function of time and temperature. DSCs are described in xe2x80x9cDifferential Scanning Calorimetry: an Introduction for Practitioners,xe2x80x9d G. Hxc3x6hne, W. Hemminger and H. J. Flammersheim (Springer-Verlag: 1996) and xe2x80x9cThermal Analysis,xe2x80x9d Bernhard Wunderlich (Academic Press, 1990).
Power compensation DSCs are a specific type of DSC that measure the difference in power supplied to the sample compared to the reference throughout the analysis. Power compensation DSCs are described in xe2x80x9cA Differential Scanning Calorimeter for Quantitative Differential Thermal Analysisxe2x80x9d, E. S. Watson and M. J. O""Neill, Analytical Chemistry Vol. 36, No 7, pp. 1233-1238 (June 1964), xe2x80x9cThe Analysis of a Temperature-Controlled Scanning Calorimeterxe2x80x9d, M. J. O""Neill, Analytical Chemistry Vol. 36, No 7, pp. 1238-1245 (June 1964), and in U.S. Pat. No. 3,263,484 to Watson and O""Neill and U.S. Pat. No. 3,732,722 to Norem, O""Neill, and Richmond, which are incorporated by reference herein.
A sample to be analyzed and a reference are heated (or cooled) in two independent sample holders. Each of the sample holders incorporates a detector to measure the temperature of the sample holder and a heating element to heat the sample holder. The sample holders are heated (or cooled) so that the average temperature of the sample holders follows the desired temperature program. Because of the presence of a sample, the temperature of the two sample holders should diverge as the average temperature of the sample holders is increased (or decreased). Typically, the sample would heat (or cool) more slowly and the reference would heat (or cool) more rapidly than the programmed heating (or cooling) rate. To prevent this from occurring, a proportional controller regulates the power supplied to the sample holder and to the reference holder. The power supplied to the sample holder is increased (or decreased) by a small amount and the power supplied to the reference is decreased (or increased) by the same amount so that the temperature difference between the sample and reference is controlled. This small differential power is approximately equal to the heat flow to the sample and is the measured heat flow.
FIG. 1 shows a thermal network model that may be used to represent certain configurations of power compensation DSCs. T0 is the temperature of the isothermal enclosure surrounding the sample holders, Ts is the temperature of the sample holder and Tr is the temperature of the reference holder. Rs and Rr represent the thermal resistance of the sample and reference portions of the calorimeter, respectively. Cs and Cr represent the thermal capacitance of the sample and reference portions of the calorimeter, respectively. Thermal capacitance is the product of mass and specific heat and is a measure of the heat storage capacity of a body. The heat flow to the sample and the heat flow to the reference and their pans are represented by qs and qr, respectively. The power supplied to the sample and the power supplied to the reference holders are represented by ps and pr, respectively. During the execution of a thermal program the temperature of the isothermal enclosure T0 is constant. Sample and reference powers ps and pr are applied to the sample and reference holders to maintain the average heating rate and to control the difference in temperature between the sample and reference holders. Performing a heat balance on the sample holder yields:                               q          s                =                                                            T                o                            -                              T                s                                                    R              s                                +                      p            s                    -                                    C              s                        ·                                          ⅆ                                  T                  s                                                            d                τ                                                                        (        1        )            
Similarly, a heat balance on the reference holder gives,                               q          r                =                                                            T                o                            -                              T                r                                                    R              r                                +                      p            r                    -                                    C              r                        ·                                          ⅆ                                  T                  r                                                            d                τ                                                                        (        2        )            
The desired quantity is the difference between the sample and reference heat flows:
q=qsxe2x88x92qrxe2x80x83xe2x80x83(3)
Substituting for qs and qr yields:                     q        =                              p            s                    -                      p            r                    +                                                    T                o                            -                              T                s                                                    R              s                                -                                                    T                o                            -                              T                r                                                    R              r                                -                                    C              s                        ·                                          ⅆ                                  T                  s                                                            d                τ                                              +                                    C              r                        ·                                          ⅆ                                  T                  r                                                            d                τ                                                                        (        4        )            
Substitute the following expressions into the heat flow equation:
xcex94T=Tsxe2x88x92Trxcex94T0=T0xe2x88x92Tsxcex94ps=psxe2x88x92prxe2x80x83xe2x80x83(5)
The result is the power compensation DSC heat flow equation:                     q        =                              Δ            ⁢                          xe2x80x83                        ⁢            p                    +                      Δ            ⁢                          xe2x80x83                        ⁢                                          T                o                            ·                              (                                                                            R                      r                                        -                                          R                      s                                                                                                  R                      r                                        ·                                          R                      s                                                                      )                                              -                                    Δ              ⁢                              xe2x80x83                            ⁢              T                                      R              r                                +                                    (                                                C                  r                                -                                  C                  s                                            )                        ·                                          ⅆ                                  T                  s                                                            d                τ                                              -                                    C              r                        ·                                                            ⅆ                  Δ                                ⁢                                  xe2x80x83                                ⁢                T                                            d                τ                                                                        (        6        )            
This equation has five terms. The first term is the difference in power supplied to the sample position versus the power supplied to the reference position. The second term accounts for differences between the thermal resistances of the sample and reference holders. The third term accounts for the heat flow that results from the difference in temperature between the sample and reference. The fourth term is the heat flow resulting from imbalances in thermal capacitance between the sample and reference holders. The fifth term reflects heat flow resulting from differences in heating rate between the sample and reference holders. In the prior art, this equation is not used; instead a very simplified equation is used:
q=Kxc2x7xcex94Txe2x80x83xe2x80x83(7)
where K is a temperature dependent proportionality factor. This equation does not include the effects of imbalances between the sample and reference holders (the second and fourth terms in the heat flow equation), nor does it include the fifth term which expresses the differences in heating rate between the sample and reference holders. In essence, in the prior art it is assumed that the DSC is perfectly balanced, i.e., that Rs=Rr and that Cs=Cr. In reality, because of manufacturing imprecision and the variability of the heat exchange processes between the sample holder and the isothermal enclosure and between the reference holder and the isothermal enclosure, imbalances generally exist. These imbalances contribute to baseline heat flow deviations that may be significant.
The fifth term is generally very nearly equal to zero, except when a transition is occurring in the sample, for instance during a melt. Usually the transition heat flow signal is integrated over a suitable baseline to obtain the total energy of the transition. Because the integral of the fifth term over the transition is zero, it is conventionally ignored in the prior art. However, it may contribute significantly to the shape of the heat flow curve during a transition. Thus, by including the fifth term, the dynamic response of the instrument is improved. Also, as noted by Hxc3x6hne et. al., referenced above, this term must be taken into account when a partial integration of the transition peak is performed (for instance when kinetic investigations to determine purity are undertaken). When the fifth term is included, the return to baseline after the completion of a transition is more rapid. Because the resolution of a DSC is its ability to separate transitions that occur in a sample within a small temperature interval, and that ability is determined solely by how quickly the heat flow signal decays after a transition is complete, including the fifth term of the DSC heat flow equation improves the resolution of the DSC by increasing the rate of decay of the heat flow signal after a transition is completed.
The present invention is a power compensation differential scanning calorimeter that uses two differential temperature measurements and a five term heat flow equation to model the instrument. The present invention is also a method by which the thermal parameters required to apply the five term heat flow equation are determined. Differential scanning calorimeters employing this invention will have empty DSC cell heat flow that is much closer to zero (leading to improved baselines) and will have substantially improved resolution over conventional instruments.
In a preferred embodiment, the two differential temperature measurements are the differential temperature xcex94T0 across thermal resistance Rs, and the differential temperature xcex94T between the sample and reference holders. The absolute temperature of the sample holder and the power difference between the sample and reference holders are also measured (i.e., the differential power to the sample with respect to the reference). Additionally, the four thermal parameters, Rs, Rr, Cs and Cr must be known. The use of two differential temperature measurements allows the use of a heat flow model that includes all five terms of the five term heat flow equation. The heat flow signal that results has improved baseline performance and improved dynamic response. In particular, because the heat flow signal is much greater during a melt, the calorimeter has greater sensitivity during the melt.
Other choices of the two differential temperature measurements are also suitable, as explained below.
The present invention also comprises a method by which the four thermal parameters Cs, Cr, Rs, Rr are determined. This determination constitutes heat flow calibration of the DSC.
Heat flow calibration requires two experiments from which the four thermal parameters can be calculated. The first experiment is performed with an empty DSC cell. The DSC program begins with an isothermal temperature segment at a temperature below the lowest temperature of the desired calibration range, followed by a constant heating rate temperature ramp, and ending with an isothermal temperature segment above the highest temperature of the desired calibration range. The heating rate should be the same as the heating rate that is to be used for subsequent experiments. The second calibration experiment is performed with sapphire samples loaded in both the sample and reference holders. The same thermal program is used for the second experiment as was used for the first (empty DSC) experiment. The two calibration experiments and the calculation of the thermal parameters based on the experiments are explained in detail below.
It is the object of the present invention to provide a power compensation DSC that uses two differential temperature measurements, one absolute temperature measurement and a differential power measurement, so that a five term heat flow equation may be used to calculate a more complete and correct measure of the heat flow to the sample.
Another object of the present invention is to disclose a method by which the detector parameters needed to employ the five term heat flow equation may be determined.
A further object of the present invention is to provide a power compensation DSC with improved baseline heat flow and improved dynamic response.